When the first images of the Trinity nuclear bomb test were revealed to the public, physicists not involved with the project were able to approximate the energy of the bomb (very top-secret information!) based just on the images. Actually, it was a series of images, with a scale bar indicating size, and each image was timestamped. That is: the explosion radius, $R$, was known as a function of time $t$.
Additionally relevant to the explosion was the energy of the bomb, $E$, and the density of the air in front of the shock wave, $\rho$. The physicists were able to analyze the problem in terms of dimensional analysis!
Part a: Using dimensional analysis, write out the dependence of the explosion radius $R$, on the other parameters in the problem. You only need to consider the dimensionality, not any numerical prefactors.
Part b: At a secret location, students and instructors of MCEN 3021 have gathered with some explosives for science. We know the amount of energy in our explosives is $E_1$, and at a certain time $t^*$ after ignition, the radius of our explosion is $R_1$. From the images of the nuclear test, we discern the radius at $t^*$ was $R_2$. Write an equation for the strength of the nuclear explosion, $E_2$, in terms of the above variables. You may assume the air density at our location was the same as the density at the Trinity site.
